We investigate the average behavior of the $n$th normalized Fourier coefficients of the $j$th ($j\ge 2$ be any fixed integer) symmetric power $L$-function (i.e., $L(s,{\mathrm{sym}}^{j}f)$), attached to a primitive holomorphic cusp form $f$ of weight $k$ for the full modular group $SL(2,\mathbb{Z})$ over certain sequences of positive integers. Precisely, we prove an asymptotic formula with an error term for the sum $$S_j^{*}:=\sum _{a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\le x(a_1,a_2,a_3,a_4,a_5,a_6)\in {\mathbb{Z}}^6}{\lambda }_{{\mathrm{sym}}^{j}f}^2(a_1^2+a_2^2+a_3^2+a_4^2+a_5^2+a_6^2),$$ where $x$ is sufficiently large, and $$L(s,{\mathrm{sym}}^{j}f):=\sum _{n=1}^{\infty }\frac{{\lambda }_{{\mathrm{sym}}^{j}f}\left(n\right)}{n^s}.$$ When $j=2$, the error term which we obtain improves the earlier known result.

Let $E/F$ be a modular elliptic curve defined over a totally real number field $F$ and let $\phi$ be its associated eigenform. This paper presents a new method, inspired by a recent work of Bertolini and Darmon, to control the rank of $E$ over suitable quadratic imaginary extensions $K/F$. In particular, this argument can also be applied to the cases not covered by the work of Kolyvagin and Logachëv, that is, when $[F:\mathbb{Q}]$ is even and $\phi$ not new at any prime.

Let $f$ be a non-CM newform of weight $k\ge 2$. Let $L$ be a subfield of the coefficient field of $f$. We completely settle the question of the density of the set of primes $p$ such that the $p$-th coefficient of $f$ generates the field $L$. This density is determined by the inner twists of $f$. As a particular case, we obtain that in the absence of nontrivial inner twists, the density is $1$ for $L$ equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions...

Let ${\lambda }_f\left(n\right)$ be the nth normalized Fourier coefficient of a holomorphic or Maass cusp form f for SL(2,ℤ). We establish the asymptotic formula for the summatory function ${\sum }_{\begin{array}{c}n\le x\\ n\equiv l\left(modq\right)\end{array}}{\left|{\lambda }_f\left(n\right)\right|}^{2j}$ as x → ∞, where q grows with x in a definite way and j = 2,3,4.

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma =\mathrm{SL}(2,\mathbb{Z})$. Denote by ${\lambda }_f\left(n\right)$ the $n$th normalized Fourier coefficient of $f$. We are interested in the average behaviour of the sum $$\sum _{a^2+b^2\le x}{\lambda }_f^j(a^2+b^2)$$ for $x\ge 1$, where $a,b\in \mathbb{Z}$ and $j\ge 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$-functions and Rankin-Selberg $L$-functions.

Nous rappelons que Manin décrit l’homologie singulière relative aux pointes de la courbe modulaire $X_0\left(N\right)$ comme un quotient du groupe ${\mathbf{Z}}^{\left({\mathbf{P}}^1\right(\mathbf{Z}/N\mathbf{Z}\left)\right)}$. En s’appuyant sur des techniques de fractions continues, nous donnons une expression indépendante de $N$ d’un relèvement de l’action des opérateurs de Hecke de $H_1(X_0\left(N\right),ptes,\mathbf{Z})$ sur ${\mathbf{Z}}^{\left({\mathbf{P}}^1\right(\mathbf{Z}/N\mathbf{Z}\left)\right)}$.